In the notes "a Brief history of elliptic integral addition theorems", the author states at the very beginning of Chapter 4 that Euler found a general solution of the equation $$ \frac{dx}{\sqrt{1-x^4}}=\frac{dy}{\sqrt{1-y^4}}\tag{1} $$ to be
$$ x=\frac{c\sqrt{1-y^4}+y\sqrt{1-c^4}}{1+x^2c^2}\tag{2} $$
I don't have a problem proving that this is a solution, but how could Euler have possibly arrived at such solution?